論文 - 岩尾　慎介

Tetrahedron equation and Schur functions

Shinsuke Iwao, Kohei Motegi, Ryo Ohkawa

Journal of Physics A: Mathematical and Theoretical 2024年11月

査読有り

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Closed k-Schur Katalan functions as K-homology Schubert representatives of the affine Grassmannian

Takeshi Ikeda, Shinsuke Iwao, Satoshi Naito

Transactions of the American Mathematical Society, Series B 2024年03月

査読有り,  ISSN  2330-0000

　概要を見る

<p>Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theoretic <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Schur Katalan functions is identified with the Schubert structure sheaves in the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-homology of the affine Grassmannian. Our main result is a proof of this conjecture.</p>

<p>We also study a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theory ring of the flag variety to a closed <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.</p>

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Free-fermions and canonical Grothendieck polynomials

Shinsuke Iwao, Kohei Motegi, Travis Scrimshaw

Algebraic Combinatorics 7 （ 1 ） 245 - 274 2024年02月

査読有り,  ISSN  2589-5486

　概要を見る

We give a presentation of refined (dual) canonical Grothendieck polynomials
and their skew versions using free-fermions. Using this, we derive a number of
identities, including the skew Cauchy identities, branching rules, expansion
formulas, and integral formulas.


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Free fermions and Schur expansions of multi-Schur functions

Shinsuke Iwao

Journal of Combinatorial Theory, Series A, Volume 198, 2023 198 2023年08月

査読有り,  ISSN  0097-3165

　概要を見る

Multi-Schur functions are symmetric functions that generalize the
supersymmetric Schur functions, the flagged Schur functions, and the refined
dual Grothendieck functions, which have been intensively studied by Lascoux. In
this paper, we give a new free-fermionic presentation of them. The multi-Schur
functions are indexed by a partition and two ``tuples of tuples'' of
indeterminates. We construct a family of linear bases of the fermionic Fock
space that are indexed by such data and prove that they correspond to the
multi-Schur functions through the boson-fermion correspondence. By focusing on
some special bases, which we call refined bases, we give a straightforward
method of expanding a multi-Schur function in the refined dual Grothendieck
polynomials. We also present a sufficient condition for a multi-Schur function
to have its Hall-dual function in the completed ring of symmetric functions.


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Free-fermions and skew stable Grothendieck polynomials

Shinsuke Iwao

Journal of Algebraic Combinatorics （Springer Science and Business Media {LLC}）  56 （ 2 ） 493 - 526 2022年04月

査読有り,  ISSN  0925-9899

　概要を見る

Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew
Schur polynomials. We give a free-fermionic presentation of skew stable
Grothendieck polynomials and their dual symmetric functions. By using our
presentation, we derive a family of determinantal formulas, which are
$K$-analogues of the Jacobi-Trudi formula for skew Schur functions. We also
introduce a combinatorial method to calculate certain expansions of skew (dual)
stable Grothendieck polynomials by using the non-commutative supersymmetric
Schur functions.


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Neutral-fermionic presentation of the K-theoretic Q-function

Shinsuke Iwao

Journal of Algebraic Combinatorics （Springer Science and Business Media {LLC}）  55 （ 2 ） 629 - 662 2022年03月

査読有り,  ISSN  0925-9899

　概要を見る

We show a new neutral-fermionic presentation of Ikeda-Naruse's $K$-theoretic
$Q$-functions, which represent a Schubert class in the $K$-theory of coherent
sheaves on the Lagrangian Grassmannian. Our presentation provides a simple
description and yields straightforward proof of two types of Pfaffian formulas
for them. We present a dual space of $G\Gamma$, the vector space generated by
all $K$-theoretic $Q$-functions, by constructing a non-degenerate bilinear form
that is compatible with the neutral fermionic presentation. We give a new
family of dual $K$-theoretic $Q$-functions, their neutral-fermionic
presentations, and Pfaffian formulas.


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Grothendieck polynomials and the boson-fermion correspondence

Shinsuke Iwao

Algebraic Combinatorics （Cellule {MathDoc}/{CEDRAM}）  3 （ 5 ） 1023 - 1040 2020年10月

査読有り,  ISSN  2589-5486

　概要を見る

In this paper we study algebraic and combinatorial properties of Grothendieck
polynomials and their dual polynomials by means of the Boson-Fermion
correspondence. We show that these symmetric functions can be expressed as a
vacuum expectation value of some operator that is written in terms of
free-fermions. By using the free-fermionic expressions, we obtain alternative
proofs of determinantal formulas and Pieri type formulas.


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Jeu de taquin, uniqueness of rectification and ultradiscrete KP

Shinsuke Iwao

Journal of Integrable Systems （Oxford University Press (OUP)）  4 （ 1 ）  2019年01月

査読有り,  ISSN  2058-5985

　概要を見る

<jats:title>Abstract</jats:title>
<jats:p>In this article, we study tropical-theoretic aspects of the ‘rectification algorithm’ on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi–Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama–Kakei.</jats:p>

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Peterson Isomorphism in K-theory and Relativistic Toda Lattice

Takeshi Ikeda and Shinsuke Iwao and Toshiaki Maeno

International Mathematics Research Notices （Oxford University Press ({OUP})）  2020 （ 19 ） 6421 - 6462 2018年04月

査読有り,  ISSN  1073-7928

　概要を見る

<title>Abstract</title>
The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

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Tropical integrable systems and Young tableaux: shape equivalence and Littlewood–Richardson correspondence

Shinsuke Iwao

Journal of Integrable Systems （Oxford University Press ({OUP})）  3 （ 1 ）  2018年01月

査読有り,  ISSN  2058-5985

　概要を見る

We present a new characterization of the shape equivalent class and the
Littlewood-Richardson correspondence of Young tableaux in terms of tropical
(ultradiscrete) integrable systems. As an application, an alternative proof of
the "shape change theorem" is given.


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The discrete Toda equation revisited: Dual β-Grothendieck polynomials, ultradiscretization, and static solitons

Iwao, S. and Nagai, H.

Journal of Physics A: Mathematical and Theoretical （{IOP} Publishing）  51 （ 13 ）  2018年

査読有り,  ISSN  1751-8121

　概要を見る

This paper presents a study of the discrete Toda equation
$(\tau_n^t)^2+\tau_{n-1}^t\tau_{n+1}^t=\tau_n^{t-1}\tau_n^{t+1}$, that was
introduced in 1977. In this paper, it has been proved that the algebraic
solution of the discrete Toda equation, obtained via the Lax formalism, is
naturally related to the dual Grothendieck polynomial, which is a $K$-theoretic
generalization of the Schur polynomial. A tropical permanent solution to the
ultradiscrete Toda equation has also been derived. The proposed method gives a
tropical algebraic representation of static solitons. Lastly, a new cellular
automaton realization of the ultradiscrete Toda equation has been proposed.


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Peterson Isomorphism in $K$-theory and Relativistic Toda Lattice

Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno

2017年03月

　概要を見る

The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by
Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf
subring of the ring of symmetric functions. On the other hand, for the quantum
$K$-theory of the flag variety $Fl_n$, Kirillov and Maeno provided a
conjectural presentation based on the results obtained by Givental and Lee. We
construct an explicit birational morphism between the spectrums of these two
rings. Our method relies on Ruijsenaars's relativistic Toda lattice with
unipotent initial condition. From this result, we obtain a $K$-theory analogue
of the so-called Peterson isomorphism for (co)homology. We provide a conjecture
on the detailed relationship between the Schubert bases, and, in particular, we
determine the image of Lenart--Maeno's quantum Grothendieck polynomial
associated with a Grassmannian permutation.


Tropical Krichever construction for the non-periodic box and ball system

Shinsuke Iwao and Hidetomo Nagai and Shin Isojima

Journal of Physics A: Mathematical and Theoretical （{IOP} Publishing）  45 （ 39 ） 395202 2012年09月

査読有り,  ISSN  1751-8113

　概要を見る

A solution for an initial value problem of the box and ball system is
constructed from a solution of the periodic box and ball system. The
construction is done through a specific limiting process based on the theory of
tropical geometry. This method gives a tropical analogue of the Krichever
construction, which is an algebro-geometric method to construct exact solutions
to integrable systems, for the non-periodic system.


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Tropical curve and integrable piecewise linear map

Rei Inoue and Shinsuke Iwao

Tropical Geometry and Integrable Systems （American Mathematical Society）  580   21 - 39 2012年

査読有り,  ISSN  0271-4132

　概要を見る

We present applications of tropical geometry to some integrable
piecewise-linear maps, based on the lecture given by one of the authors (R. I.)
at the workshop "Tropical Geometry and Integrable Systems" (University of
Glasgow, July 2011), and on some new results obtained afterward. After a brief
review on tropical curve theory, we study the spectral curves and the isolevel
sets of the tropical periodic Toda lattice and the periodic Box-ball system.


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The two-dimensional periodic box-ball system and its fundamental cycle

Iwao, S.

Journal of Physics A: Mathematical and Theoretical （{IOP} Publishing）  45 （ 39 ） 395204 2012年

査読有り,  ISSN  1751-8113

　概要を見る

We study a 2-dimensional Box-Ball system which is a ultradiscrete analog of
the discrete KP equation. We construct an algorithm to calculate the
fundamental cycle, which is an important conserved quantity of the 2-dim.
Box-Ball system with periodic boundary condition, by using the tropical curve
theory.


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TROPICAL SPECTRAL CURVES, FAY'S TRISECANT IDENTITY, AND GENERALIZED ULTRADISCRETE TODA LATTICE

REI INOUE and SHINSUKE IWAO

New Trends in Quantum Integrable Systems （{WORLD} {SCIENTIFIC}）     101 - 116 2010年10月

査読有り

　概要を見る

We study the generalized ultradiscrete periodic Toda lattice T(M,N) which has
tropical spectral curve. We introduce a tropical analogue of Fay's trisecant
identity, and apply it to construct a general solution to T(M,N).


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Solution of the generalized periodic discrete Toda equation II: theta function solution

Shinsuke Iwao

Journal of Physics A: Mathematical and Theoretical （{IOP} Publishing）  43 （ 15 ） 155208 2010年03月

査読有り,  ISSN  1751-8113

　概要を見る

We construct the theta function solution to the initial value problem for the
generalized periodic discrete Toda equation.


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Integration over Tropical Plane Curves and Ultradiscretization

Iwao, Shinsuke

International Mathematics Research Notices （OXFORD UNIV PRESS）  2010 （ 1 ） 112 - 148 2010年

査読有り,  ISSN  1073-7928

　概要を見る

In this article we study holomorphic integrals on tropical plane curves in
view of ultradiscretization. We prove that the lattice integrals over tropical
curves can be obtained as a certain limit of complex integrals over Riemannian
surfaces.


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Solution to the initial value problem of the ultradiscrete periodic Toda equation

Idzumi, M. and Iwao, S. and Mada, J. and Tokihiro, T.

Journal of Physics A: Mathematical and Theoretical （IOP PUBLISHING LTD）  42 （ 31 ） 315209 2009年

査読有り,  ISSN  1751-8113

　概要を見る

We present an expression for the solution to the initial value problem for the ultradiscrete periodic Toda equation. The expression provides explicit forms of all dependent variables of the equation, while the previously known solutions give only half of the dependent variables while the others have to be determined implicitly using the conserved quantities.

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Solutions of the generalized periodic discrete Toda equation

Iwao, S.

Journal of Physics A: Mathematical and Theoretical （IOP PUBLISHING LTD）  41 （ 11 ） 115201 2008年

査読有り,  ISSN  1751-8113

　概要を見る

A box-ball system with more than one kind of balls is obtained by the
generalized periodic discrete Toda equation (pd Toda eq.). We study the pd Toda
equation in view of algebraic geometry. The time evolution of pd Toda eq. is
linearized on an algebraic variety, and theta function solutions are obtained.


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